Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?

I have seen many papers on Liouville's equation $\Delta u=K e^{ u}$ when $K>0$, such as enter link description here

or the theorem

**If $\bar{M}$ is a connected, compact 2-manifold with nonempty boundary $\partial M, g$ a Riemannian metric on $\bar{M}$, and $K \in C^{\infty}(\bar{M})$ a given function satisfying
$$
K(x) \leq 0 \text { on } M,
$$
then there exists $u \in C^{\infty}(\bar{M})$ such that the metric $g^{\prime}=e^{2 u} g$ conformal to $g$ has Gauss curvature $K$. Given any $v \in C^{\infty}(\partial M)$, there is a unique such $\mathrm{u}$ satisfying $u=v$ on $\partial M$.**

which can be transformed to be solving a PDE in the form of $\Delta u=K e^{ u}$ when $K>0$.

But I hardly saw researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$ What's the difficulty of this and are there any researches on this problem?